The Banach-Tarski paradox [1]} shows that any two bounded subsets of \(\mathbb {R}^3\)  with nonempty interior are equidecomposable.
Von Neumann [2]} introduced amenable groups in order to explain this paradox.
This is the largest class of groups where the existence of an equidecomposition may imply the existence of a measurable equidecomposition.
Gardner made the following conjecture.
